Linear complementarity representation of piecewise linear functions

We investigate a representation of piecewise linear function containing a complementarity condition, and call it a linear complementarity representation. This representation is also known as ‘Bokh1’ or ‘state-variable representation’ in literature. We discuss two spacial types of representation, called P-representation and ULT-representation, and give an elementary proof to show that either one of these two types of representation completely characterizes any piecewise linear function. We can also demonstrate that a P-representation can be transformed into a ULT-representation, and show how it can be implemented in an algorithmic manner. Furthermore, as an application of linear complementarity representation, we will present a formula of linear complementarity representation of the Choquet integral model over a finite set, which is a model used in the area of decision-making as an aggregation function.